Struct Multidisc Optim (2018) 57:1765–1777

RESEARCH PAPER

Topology optimization of compact widebandcoaxial-to-waveguide transitions with minimum-size control

Emadeldeen Hassan1,2 ·Eddie Wadbro1 ·Linus Hägg1 ·Martin Berggren1

Received: 10 April 2017 / Revised: 8 September 2017 / Accepted: 20 October 2017 / Published online: 15 November 2017© The Author(s) 2017. This article is an open access publication

Abstract This paper presents a density-based topologyoptimization approach to design compact wideband coaxial-to-waveguide transitions. The underlying optimizationproblem shows a strong self penalization towards binarysolutions, which entails mesh-dependent designs that gener-ally exhibit poor performance. To address the self penaliza-tion issue, we develop a filtering approach that consists oftwo phases. The first phase aims to relax the self penaliza-tion by using a sequence of linear filters. The second phaserelies on nonlinear filters and aims to obtain binary solutionsand to impose minimum-size control on the final design. Wepresent results for optimizing compact transitions betweena 50-Ohm coaxial cable and a standard WR90 waveguideoperating in the X-band (8–12 GHz).

Keywords Topology optimization · Compact transition ·Nonlinear filter ·Waveguide · Coaxial cable ·Maxwellequations

! Emadeldeen Hassanemad@cs.umu.se

Eddie Wadbroeddiew@cs.umu.se

Linus Hägglinush@cs.umu.se

Martin Berggrenmartinb@cs.umu.se

1 Department of Computing Science, Umeå University,SE–901 87 Umeå, Sweden

2 Department of Electronics and Electrical Communications,Menoufia University, 32952 Menouf, Egypt

1 Introduction

Modern RF and microwave systems design is continuouslyshifting toward the use of printed circuit and surfacemountedtechnologies to facilitate mass production and achieve com-pactness. Transitions are key components of microwavecircuits and are used to match signals between transmis-sion lines that have different wave impedance, propagatingmodes, or directions of propagation. A mismatched tran-sition can have a significant impact on the overall systemefficiency and can lead to overheating of the device.

Rectangular waveguides are commonly used to feed hornantennas (Balanis 2005, ch 13), as elements in phased arrayantennas (Pellegrini et al. 2014), or in material charac-terization (Chang et al. 1997). Meanwhile, for feeding ormeasurements purposes, coaxial cables are typically used tocouple signals into/from waveguides. Coaxial cables sup-port the TEM mode and possess essentially a constantcharacteristic impedance, whereas rectangular waveguidessupport TE or TM modes and have a frequency dependentwave impedance (Pozar 2012).

Coaxial-to-rectangular waveguide transitions operatingover narrow frequency bands can be designed using elec-tric probes or magnetic loops, whose configuration dependson a few parameters that are easy to determine (Keam andWilliamson 1994; Deshpande et al. 1979; Bialkowski et al.2000). However, wideband transitions typically includecomplex, bulky 3D structures that can be complicated tomass produce (Yi et al. 2011; Bang and Ahn 2014; Takoet al. 2014). Simeoni et al. (2006) proposed a compact typeof coaxial-to-rectangular waveguide transitions that is suit-able for mass production by using printed circuit boardtechnology. However, the use of elementary design shapestogether with the requirement on compactness make the pro-posed transitions exhibit narrow frequency band operation.

https://doi.org/10.1007/s00158-017-1844-8

http://crossmark.crossref.org/dialog/?doi=10.1007/s00158-017-1844-8&domain=pdfhttp://orcid.org/0000-0002-1318-7519mailto:emad@cs.umu.semailto:eddiew@cs.umu.semailto:linush@cs.umu.semailto:martinb@cs.umu.se

1766 Emadeldeen Hassan et al.

Instead of relying on fixed elementary shapes when deter-mining the layout of the printed circuit board, we will hereapply the material distribution (also called density-based)technique of topology optimization to design compact wide-band coaxial-to-rectangular waveguide transitions.

During the last decade, topology optimization has startedto be applied to the design of various electromagnetic com-ponents, such as antennas (Nomura et al. 2007; Erentokand Sigmund 2011; Zhou et al. 2010; Aage 2011; Hassanet al. 2014b), metamaterials (Diaz and Sigmund 2010; Oto-mori et al. 2012), and filters (Kiziltas et al. 2004; Nomuraet al. 2013; Aage and Egede Johansen 2017). As opposedto classical topology optimization approaches applied toelastic material, the layout optimization of conducting mate-rials suffers from an ohmic barrier issue. This phenomenonstems from the fact that a material with zero (a dielec-tric) or infinite (a metal) conductivity exhibits no ohmiclosses, in contrast to materials with intermediate conductiv-ities. When topology optimization is applied to maximizetransmission or reception, the algorithm will quickly driveeach material point to either maximum or minimum conduc-tivity values in order to minimize the losses, if no specialaction is taken. Moreover, it will be difficult for a con-tinuous optimization algorithm to change a material pointfrom metal to dielectric, or vice versa, because of the bar-rier of the intermediate conductivity values. Another wayof expressing the same issue is to say that the problem isself-penalized toward pure dielectric–metal designs. A naivegradient-based topology optimization implementation willlead to a quick convergence of the algorithm to a loss-less design, unfortunately with bad performance (Hassanet al. 2014a). This problem can be addressed by densityfiltering, a tool originally developed to regularize classi-cal topology optimization problems (Bendsøe and Sigmund2003). In previous works, we have devised a strategy forself-penalized problems based on filtering together with acontinuation approach for a decreasing filter radius. We suc-cessfully applied this approach first to the design of metallicantennas (Hassan et al. 2014a, b, 2015a). A similar use offiltering as a strategy to combat the self penalization is alsosuggested by Aage and Egede Johansen (2017).

In the initial stage of such a strategy, when the filterradius is large, the algorithm operates on a design with largeareas of material with intermediate conductivities, and asthe filter radius is decreased, intermediate conductivity val-ues are driven closer to the extreme values due to the selfpenalization. In the end, when the filter radius vanishes,almost all material points contain either a low or a highconductivity material. Note that in this strategy, there is noinherent size control of metal or etched areas, since the fil-ter radius successively vanishes. Nevertheless, in a recentwork, we successfully used this strategy to design the layoutof metal on a printed circuit board serving as the radiating

element in a coaxial-to-waveguide transition (Hassan et al.2017). In that study, the circuit board was positioned cen-tered and in-line with the extension of the waveguide. Thisposition is electrically favorable, since the electric field forthe dominant mode has a maximum in the center of thewaveguide, and the devices we obtained indeed exhibit verywide-band operational ranges. However, the components arenot compact due to the position of the circuit board.

A much smaller component is accomplished by posi-tioning the circuit board parallel and close to one end ofthe waveguide (see Fig. 1). This configuration is muchmore challenging to design, since the device is almost shortcircuited due to its closeness to the metallic back wall.When we attempted to design such a configuration usingthe continuation strategy that was successful in the in-lineconfiguration (Hassan et al. 2017), the algorithm’s lack ofcontrol over feature sizes, both for the metallic and non-metallic (etched) parts, became apparent; the final designcontained pronounced areas of scattered material and holes.Such features can increase the ohmic losses in microwavecircuits, especially when appearing at the boundaries of thedevice (Pozar 2012, §2.7), and they can also complicatethe manufacturing. It became thus necessary to improve thealgorithm to control the feature size of the metallic and thenon-metallic, etched parts. Here we present this improvedapproach, which allows the design of metal and non-metal regions with minimum-size control, and show how ithas been successfully used to design compact coaxial-to-waveguide transitions. This scheme operates in two phases.The first phase aims mainly to counteract the self penaliza-tion issue by filtering the design variables using a standardlinear density filter combined with a continuation approachover a successively reduced filter size—but only down toa predefined finite size corresponding to the selected fea-ture size. The second phase uses a nonlinear filtering ofthe design variables, based on a sequence of parameterizedharmonic mean filters (Svanberg and Svärd 2013), whichin the limit yields binary designs, while at the same time

Fig. 1 A 50-Ohm coaxial cable is connected to the rear end of an a×brectangular waveguide. The design domain ! is used for distributing aconductivity to match the two sides

Topology optimization of compact wideband coaxial-to-waveguide transitions... 1767

maintaining the minimum feature size imposed in the firstphase.

2 Problem statement

Figure 1 shows the problem setup. A coaxial cable is con-nected to one end of an a × b rectangular waveguide. In theelectromagnetic community, this setup is commonly calledend-launcher transition. We assume that, within the fre-quency band of interest, the coaxial cable supports only theTEM mode of propagation and has a constant 50 Ohm char-acteristic impedance. Generally, rectangular waveguideshave a frequency dependent wave impedance and supportdifferent modes of propagation (Pozar 2012). In this work,the dominant TE10 mode of the rectangular waveguide isconsidered as the mode of interest. The boundaries of thecoaxial cable and the waveguide are assumed to be per-fect conductors. Inside the design domain !, see Fig. 1,we aim to distribute material with conductivity σ to max-imize the transmission of signals between the waveguideand the coaxial cable. The domain ! is backed by a thin,low-loss dielectric substrate, occupying the volume between! and the shorting wall of the waveguide at z = 0, tohold the resulting conductivity distribution. The other endsof the coaxial cable and the waveguide are assumed to bematched. That is, the outgoing signals do not reflect back tothe analysis domain.

3 Governing equations and discretization

The electric field E and the magnetic field H, insidethe waveguide and the coaxial cable, are governed byMaxwell’s equations

∂tµH+ ∇ × E = 0, (1)∂tϵE+ σE − ∇ × H = 0, (2)

whereµ, ϵ, and σ are the medium permeability, permittivity,and conductivity, respectively. Under the assumption thatonly the TEM mode is supported by the coaxial cable, thepotential difference V and the current I inside the coaxialcable satisfy the following transport equation (Hassan et al.2014b):

∂t (V ± ZcI )± c∂z(V ± ZcI ) = 0 ∀t and ∀z, (3)V + ZcI = g(t) at z = z0. (4)

Here Zc and c are the characteristic impedance and phasevelocity inside the coaxial cable, respectively. The term(V + ZcI ) indicates a signal traveling in the positive z-direction and is used to impose the boundary condition atthe coaxial cable end z0, and g(t) is chosen to determine

the energy spectral density of the imposed signal. The term(V −ZcI) represents an outgoing signal traveling in the neg-ative z-direction. We use this term to estimate the outgoingenergy through the coaxial cable end by

Wout,coax =1

4Zc

∫ T

0(V − ZcI )2 dt, at z0 (5)

where T is the total simulation time.We use the finite-difference time-domain (FDTD)

method (Taflove and Hagness 2005) to numerically solvethe governing equations (1)–(4). The computational domainis discretized by a uniform cubical Yee grid, where the con-ductivity is located at the edges. The incoming energy isimposed through the waveguide by the total-field scattered-field technique. The incoming energy can also be providedby specifying a nonzero function g(t) in boundary condi-tion (4). To simulate the matched ports, the waveguide isterminated by a perfectly matched layer, and (3) togetherwith boundary condition (4) provide perfect absorption ofoutgoing waves through the coaxial cable end. To controlthe frequency spectrum of the incoming energy, we use, asa feeding signal, a truncated time-domain sinc pulse modu-lated to the center of the frequency band of interest (Lathi1998). The FDTD code is implemented to run on graph-ics processing units (GPUs) using the parallel computingplatform CUDA (https://developer.nvidia.com/what-cuda).

4 Topology optimization problem

We first consider the case when the coaxial-to-waveguidesystem is fed through the waveguide by a time-domainsignal that covers the frequency band of interest. For thisproblem we have, as illustrated in Fig. 1, the followingenergy balance

Win,wg = Wout,coax +Wout,wg +W!, (6)where Win,wg is the incoming energy associated with theTE10 mode imposed in the waveguide, Wout,wg is the out-going energy through the waveguide, and W! is ohmiclosses inside the design domain !. To obtain a good transi-tion between the waveguide and the coaxial cable, a naturalobjective is to maximize the outgoing energy through thecoaxial cable, Wout,coax, for the given incoming energythrough the waveguide, Win,wg. According to energy bal-ance (6), the maximization of Wout,coax is equivalent to theminimization of the sumWout,wg +W!.

In our initial experiments for this project, and also in a pre-vious work (Hassan et al. 2015a), we noticed that maxi-mizing the quotientWout,coax/Wout,wg usually results in opti-mized designs with favorable performance compared tosolelymaximizingWout,coax. (MaximizingWout,coax/Wout,wgeffectively corresponds tomaximizing the transmission term

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1768 Emadeldeen Hassan et al.

Wout,coax and minimizing the reflection term Wout,wg.) So,to maximize the matching between the waveguide and thecoaxial cable, one possibility would be to formulate theconceptual optimization problem

maximize log(Wout,coax

Wout,wg

)(7)

subject to governing equations (1)–(4) and the boundaryconditions discussed in the previous section. The perfor-mance of microwave devices is typically evaluated using thescattering parameters, which are computed as the logarithmof power ratios. Thus it is natural to use the logarithmicfunction in the objective function.

As mentioned in the introduction, the waveguide sup-ports different modes of propagation and possesses a fre-quency dependent wave impedance. In time-domain simu-lations, these frequency dependencies require complicatedtreatments regarding the splitting of transient waves andimposing the absorbing boundary condition for the differ-ent modes in the rectangular waveguide (Kristensson 1993;Akgun and Tretyakov 2015). To circumvent these compli-cations, we use the fact that the problem under investigationis reciprocal (Pozar 2012). That is, in problem (7), insteadof observing Wout,wg given an incoming energy through thewaveguide, we observe Wout,coax given an incoming energythrough the coaxial cable. We carry out two simulations,one where Win,wg is nonzero and g(t) = 0, and one whereWin,wg = 0 and g(t) is nonzero. Moreover, we rewrite theconceptual problem (7) as

maximize log

(Wout,coax

∣∣Win,wg

Wout,coax∣∣Win,coax

)

(8)

subject to the governing equations and the boundary con-ditions discussed in the previous section. In problem (8),Wout,coax

∣∣Win,wg and Wout,coax

∣∣Win,coax represent the outgo-

ing energy in the coaxial cable given the incoming energythrough the waveguide and the coaxial cable, respectively.By this treatment, we are able to use the perfectly matchedlayer inside the waveguide to simulate a matched portthat works for all modes. Moreover, because of transportequation (3), the splitting of a transient wave inside thecoaxial cable is much simpler than inside the waveguide.The price for this treatment, however, is that we need tosolve the governing equations twice in order to evaluate theobjective function.

To solve optimization problem (8), we employ the mate-rial distribution topology optimization approach. We use amaterial indicator vector p = [p1, p2, · · · , pN ] to indicatepresence, pi = 1, or absence, pi = 0, of the conduc-tive material inside the design domain !. Our preliminarynumerical experiments showed a low sensitivity of theobjective function to conductivity values outside the range

[σmin = 10−3 S/m,σmax = 105 S/m]. Thus, we map thedesign vector p, also known as the density vector, to thephysical conductivity at the edges of the design grid using

σ = 10(8p−3)S/m. (9)To allow gradient-based optimization methods to solve opti-mization problem (8), the entries of the density vector areallowed to take values between 0 and 1. The topology opti-mization problem is to find p ∈ A = {x ∈ RN | 0 ≤ xi ≤1 ∀i} that solves problem (8). To compute the objectivefunction gradient, we use an adjoint-field method as detailedby Hassan et al. (2015b). The gradient vector is computedbased on the solutions of two adjoint systems correspond-ing to the two termsWout,coax

∣∣Win,wg andWout,coax

∣∣Win,coax in

problem formulation (8).As mentioned in the introduction, optimization problem

(8) is self-penalized toward the lossless cases (that is, towardσmin and σmax) due to the energy losses W! associatedwith the intermediate conductivities, illustrated in Fig. 2.In other words, when gradient-based optimization methodsare used to solve this problem, these methods quickly con-verge to designs consisting of the two extreme values ofthe design variables, since these values minimize the energyloss. Unfortunately, the quick convergence caused by theself penalization typically results in designs that exhibit badperformance (Hassan et al. 2014a).

5 Filtering

Filtering procedures (Sigmund 1994; Bourdin 2001; Brunsand Tortorelli 2001) are among the most popular strategiesto achieve mesh-independent designs in topology optimiza-tion. Here we consider density filtering methods, where thedesign variables are filtered and the filtered design vari-ables are mapped to the coefficients that enter the governingequation. A disadvantage of the original linear density fil-ter is that it produces designs with relatively large areas of

Fig. 2 A typical variation of the energy loss against the values of thedesign domain conductivity

Topology optimization of compact wideband coaxial-to-waveguide transitions... 1769

intermediate densities. During the last decade there has beenan increased interest in nonlinear filtering strategies, aimedat reducing the amount of intermediate densities (Guest etal. 2004, 2011; Sigmund 2007; Svanberg and Svärd 2013;Wadbro and Hägg 2015; Hägg and Wadbro 2017). In addi-tion to providing mesh independent solutions, filtering ofthe design variables can relieve the strong self penalizationdiscussed in the previous section. To compute the physicalconductivity in expression (9), we replace the density vec-tor p by the filtered density p̃ obtained by a cascade of KfW-mean filters (Wadbro and Hägg 2015),

p̃ def= F(K) ◦ F(K−1) ◦ · · · ◦ F(1)(p), (10)

where

F(k)(p) = f−1k (W(k) fk(p)) ∀k ∈ {1, · · · ,K}, (11)

and where the matrix W(k) determines the weights, shape,and the size of the kth filter; the functions fk(·) and theirinverses f−1k (·) are applied elementwise. The filter size isdefined by a variable R that defines the radius of a circu-lar neighborhood around each point. In this work, we useequal weights in each neighborhood. The formulation of thegeneralized fW-mean filter allows us to investigate differenttypes of filters based on the choice of the functions fk(·).Here we employ linear as well as nonlinear filters.

5.1 Linear filter

By explicitly setting the function fk(x) = x for all k,the cascade of fW-mean filters reduces to a linear den-sity filter. In this case, the filtering process replaces eachdesign variable pi by a weighted arithmetic mean of thedesign variables within a neighborhood defined by theweight matrixW(K)W(K−1) . . .W(1). The linear filter coun-teracts the self penalization of the optimization problem byproducing designs with intermediate values.

To obtain a binary design at the end of the optimiza-tion process, we use a continuation approach over the filterradius. In particular, we start by a large filter size Rmax andsolve a sequence of problems with decreasing filter radii,Rn = γ nRmax with γ < 1, until the filter radius dropsbelow the spatial discretization step & used in the FDTDgrid. In other words, we gradually remove the impact of thefilter. Unfortunately, this strategy produces mesh-dependentsolutions.

5.2 Non-linear filter

To pursue nonlinear filtering of the design variables, wesubstitute fk(·) in expression (11) by harmonic functionsas proposed by Svanberg and Svärd (2013) for elasticityproblems. More precisely, we use functions of the form

fk = (x+α)−1 or fk = (1−x+α)−1. The parameter α > 0is used to control the nonlinearity of fk(x). When α tendsto infinity the harmonic filters approach a linear filter. Forsmall values of α the harmonic filters allow for sharper tran-sitions between regions of different materials than the linearfilter. In addition, the harmonic filter operator have boundedderivatives as the nonlinearity parameter α tends to zero. Werefer to the study by Svanberg and Svärd (2013) for a com-prehensive comparison between the harmonic mean filtersand other kinds of filters.

The fW -mean filters with the first (second) of the har-monic functions above promotes the values 0 (1), providedthat the input contains 0 (1). The behavior of the two har-monic filters, for small values of α, is in this sense similarto the erode and dilate operators in mathematical morphol-ogy (Haralick et al. 1987; Heijmans 1995). On a binarydesign, the dilate operator dilates regions occupied withones. That is, this operator expands areas occupied by metal.The effect of the erode operator is the opposite in the sensethat it erodes regions occupied with ones; that is, it dilatesregions occupied with zeros. If we cascade an erode opera-tor followed by a dilate operator, then we get the so-calledopen operator that removes regions containing ones that aresmaller than the neighborhood. Similarly, if we cascade adilate operator followed by an erode operator, then we getthe close operator that removes regions containing zeros thatare smaller than the neighborhood.

In the numerical experiments, whenever we use non-linear filtering, we use an open–close filter; that is, anopen filter followed by a close filter. The harmonic open–close filter is thus a cascade of four filters, where f1(x) =f4(x) = (x + α)−1 and f2(x) = f3(x) = f1(1 − x).We remark that, in general, the open-close filtering strategydoes not guarantee minimum size control on both metal andetched regions, as Schevenels and Sigmund (2016) recentlyreported. Nevertheless, our numerical experiments suggestthat the results obtained using an open–close filtering strat-egy in many cases provide final designs that have smoothboundaries as well as minimum size control on both themetal and etched regions.

6 Design algorithm

Figure 3 shows a summary of the proposed optimizationalgorithm. We start with a uniform distribution of the designvariables. The first loop in the design algorithm shows thebasic steps to solve a topology optimization problem. In thisloop, we filter the design variables, compute the objectivefunction and its gradient, check a convergence criterion, andthen, if needed, update the design variables. As convergencecriterion, we use the norm of the first-order optimality con-dition relative to a reference value. More precisely, we start

1770 Emadeldeen Hassan et al.

Initial distributionInitialize filter parameters

Filtering of design variables

Compute Objective(Maxwell’s equations (FDTD))

Compute gradient(solve adjoint Maxwell’s equ.)

Filter the gradient (chain rule)

Updatedesign

variables(G

CMMA)

Converged?

Updatefilter

Final design

yes

yes

yes

no

no

no

The filter radius

The nonlinearity variable

Fig. 3 The optimization algorithm including two continuation levels;one over the filter radius and one over the nonlinearity of the filter

the solution of a problem, record the norm of the first-orderoptimality condition after 8 iterations, continue the solutionof the subproblem until the norm of the first-order optimal-ity condition decreases to 25% of the recorded value. Toupdate the design variables, we use the globally-convergentmethod of moving asymptotes (GCMMA) developed bySvanberg (2002).

The second two loops in the design algorithm update thefilter parameters. The first of these represents the continua-tion over the filter radius. In all cases, we update the filterradius using Rn = max{(0.75)n × (10&), Rmin}, where weset Rmin = & when linear filters are used. For the nonlin-ear filters, we investigate different values for Rmin. Whennonlinear filters are used, the last loop represents the con-tinuation over the nonlinearity variable α. We update thenonlinearity variable using αi = (0.5)i × αmax, where αmaxis a starting value of α chosen to be large enough. The algo-rithm terminates when αi drops below αmin = 10−6. Tocharacterize the amount of grayness in the final design, weuse the non-discreteness measure, Mnd = 4p̃T (1 − p̃)/N ,suggested by Sigmund (2007), where N is the number ofentries in p̃ and 1 is a vector of length N with ones at allentries. Furthermore, in a final post-processing step, we use1 S/m as a threshold conductivity to map values below andabove that value to σmin and σmax, respectively.

7 Results and discussion

We design a transition between a 50 Ohm coaxial cableand a standardWR90 rectangular waveguide. The frequencyband 8–12 GHz is considered as the band of interest. Theradii of the inner probe and the outer shield of the coax-ial cable are 1.27 mm and 4.45 mm, respectively, and thewaveguide dimensions are 22.86mm×10.16mm. The coax-ial cable is connected, to the shorting wall at the waveguideend, at a point shifted 2.54 mm in the negative x-directionfrom the center (see Fig. 1). We use a spatial discretiza-tion step & = 0.127 mm in the FDTD method and a timediscretization 0.95 of the Courant step. The design domain!, see Fig. 1, located at z = 1.27 mm, is discretized into180 × 80 cells, resulting in 28,540 design variables associ-ated with the interior edges of the grid. The volume betweenz = 0 and z = 1.27 mm is filled by a low-loss RT/Duroid5880 LZ substrate (ϵr = 1.96 and tan δ = 0.002 at 10 GHz)to hold the design. For all design problems in this work, westart with a uniform initial distribution of the design vari-ables, pi = 0.6, which corresponds to conductivity valuesnear the peak in Fig. 2. This choice reduces the risk of bias-ing the design towards any of the two lossless cases (that is,the good dielectric and the good conductor).

7.1 Linear filter

In this section, we present the results for designing the tran-sition using a linear filter combined with a continuationapproach. First, we set the functions fk(x) = x in expres-sion (11), and use a cascade of four linear filters (that is,K = 4 in expression (10)). The reason to use the four cas-caded linear filters is to obtain a filtering effect, especiallyin the beginning of the optimization, comparable to the oneused in the next section, for the nonlinear filter. Althougheach filter matrix W(k) uses constant weights over a circu-lar neighborhood of radius R, the cascade of the fW-meanfilters acts as a linear filter with weighting matrixW4 (filtersize 4R) (Hägg and Wadbro 2017).

Figure 4 shows the progress of the objective functiontogether with some snapshots showing the developmentof the filtered design. We include in the same figure thechange in the level of non-discreteness of the filtered design.The objective function increases monotonically with dis-tinct jumps at the beginning of each subproblem, when thefilter radius decreases. At the beginning of the optimization,the design contains large amounts of intermediate values(Mnd = 44%), and the boundaries are blurred, which isexpected because of the large filter size. At the end of theoptimization process, the design has crisp boundaries andMnd has decreased to 1.6%.

Figure 5a shows the final conductivity distributionobtained by the optimization algorithm after 111 iterations.

Topology optimization of compact wideband coaxial-to-waveguide transitions... 1771

0 20 40 60 80 100Number of iterations

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

obj

ectiv

e fun

ctio

n

0

10

20

30

40

50

Non

-disc

rete

ness

(%)

Fig. 4 The iteration history of the objective function and the non-discreteness level together with some samples of the filtered design.Black color indicates a good conductor and white color indicate a gooddielectric

We note that the final design contains small metallic partsas well as small holes inside the metallic block. These smallfeatures typically appear in the design when the filter radiusis reduced to small values, which indicates that the solu-tion to the optimization problem is mesh dependent. (Aswe use finer grids, smaller features will likely appear inthe final design.) Based on numerical investigations, thesesmall metallic parts and holes have a minor impact on theperformance of the transition. This conclusion can also beinferred from the development of the objective functionin Fig. 4; there is little variation in the objective functionnear the end, where the small features start to appear. Inpractice, however, these small features may complicate themanufacturing process and raise the ohmic losses insidethe transition, as mentioned in the introduction. Figure 5bshows the amplitudes of the reflection coefficient |S11| andthe coupling coefficient |S21| of the transition. The perfor-mance of the transition is computed by our FDTD code and

cross-verified with the commercial CST Microwave Stu-dio package (https://www.cst.com/). Overall, there is a goodmatch between the two simulations; the slight frequencyshift can be accounted for by the differences in geometrydescription between the two methods. The optimized tran-sition has a reflection coefficient, |S11|, below −8 dB andcoupling coefficient, |S21|, above −1 dB over the frequencyband 8.5–12 GHz, which essentially covers the frequencyband of interest marked by the vertically dashed lines in theplot. The simulation results of the transition show that thereis a resonance around 11.5 GHz, which also appears, unfor-tunately, in many of the subsequent optimization results. Weemphasize that the formulation of the objective function asthe integral of the outgoing energy in time-domain makesit difficult to target a specific single frequency. We believethat this resonance is promoted by some geometrical fea-tures in the problem setup and leave this issue for futureinvestigations.

We note that the final designs may be asymmetric withrespect to the symmetry plane y = a/2 (see Fig. 1).Although the problem setup is symmetric, an optimal designneed not be symmetric. In fact, imposing symmetry wouldrestrict the design space, which potentially leads to designswith poorer performance. Here, the asymmetry stems fromthe use of finite precision arithmetic; although the initialdesign conductivity is symmetric, the numerically computedfields will generally not be perfectly symmetric. Givenasymmetric design conductivities, the optimization algo-rithm may drive the design back to the symmetric case oraway from it depending on the design sensitivities.

To further improve the performance of the transition, weuse an additional design layer inside the waveguide at theplane z = 2.54 mm to distribute the conductivity. The newdesign domain consists of two layers, one layer at z = 1.27mm and the second layer at z = 2.54 mm, with a low-lossRT/Duroid 5880 LZ substrate filling the volume between

Fig. 5 a The final conductivitydistribution over the designdomain when a linear filtercombined with a continuationover the filter radius is used. bThe scattering parameters of thetransition

(a)

5 6 7 8 9 10 11 12 13 14 15Frequency (GHz)

-25

-20

-15

-10

-5

0

Am

plitu

de (d

B)

(b)

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1772 Emadeldeen Hassan et al.

(a) (b)

5 6 7 8 9 10 11 12 13 14 15Frequency (GHz)

-25

-20

-15

-10

-5

0

Am

plitu

de (d

B)

(c)Fig. 6 Results of optimizing the two-layer design using the linear filtering approach. a First layer conductivity distribution, z = 1.27 mm. bSecond layer conductivity distribution, z = 2.54 mm. c The scattering parameters of the transition

z = 0 mm and z = 2.54 mm. The new design problem has57,080 design variables and is solved using the same set-tings as the one-layer design case. Figure 6a and b show thefinal design obtained after 92 iterations. We note the appear-ance of small metallic parts in the second layer at z = 2.54mm. These small features, as mentioned for the one-layerdesign, appear when the filter size diminishes, since thesolution of the optimization problem becomes mesh depen-dent. The non-discreteness level of the final design isMnd =0.3%. Figure 6c shows the improvement in the performanceof the two-layer design compared to Fig. 5b. The |S11| curvehas values below −12.5 dB over the frequency band 8.4–12GHz with a corresponding |S21| above −0.5 dB, excludingthe resonance around 11 GHz.

To conclude, the approach used in this section has thedrawback that small features can appear in the final designs,as shown in Figs. 5a and 6b. These features appear as aconsequence of reducing the filter radius during the con-tinuation approach. These small features can increase theohmic losses (Pozar 2012, §2.7) and cause manufacturingproblems.

7.2 Nonlinear filters

In this section, we present the results when using the pro-posed two-phase filtering approach. We use a cascade offour fW-mean filters to implement the open–close filter dis-cussed in Section 5. In a one dimensional test problem,we noticed only small differences between filtered designsobtained by using values of α > 8 in the harmonic open–close (fW-mean) filter and the case when a linear filter isused (that is, when fk(x) = x for all k). Therefore, in the

first phase of the filtering process, Phase I, we fix the nonlin-earity variable α to αmax = 8, and we follow a continuationapproach over the filter radius, as in the case of the linearfilter. That is, we solve a sequence of problems, for whichthe the filter radius is updated using Rn = (0.75)n × (10&).Here, the filter radius is updated until the radius reaches aspecified value Rmin ∈ {3&, 5&, 7&}. In the second phaseof the filtering process, Phase II, we fix the filter radius toRmin and start a continuation approach over the nonlinear-ity variable α aiming for binary solutions. In Phase II, weupdate the nonlinearity variable using αi = (0.5)i×(8), andthe algorithm stops when αi drops below 10−6. We remarkthat if the required final filter size Rmin is large, then PhaseI only has a minor influence, since the harmonic open–close(fW-mean) filter performs similar to a linear filter in the

0 40 80 120 160 200 240 280 320Number of iterations

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Fig. 8 a The final conductivitydistribution over the one-layerdesign domain, when we use thetwo-phase filtering approachwith Rmin = 3&. b Thescattering parameters of thetransition

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beginning of Phase II. However, in our numerical experi-ments, we noticed that generally the use of the two-phasefiltering approach allows the algorithm to converge to bet-ter solutions in terms of the objective function. In particular,Phase I is essential to obtain good results when a small Rminis used.

Figure 7 shows the progress of the objective function andthe non-discreteness level of the filtered design for the one-layer design case using the two-phase filtering approach,when Rmin = 3& is used. The optimization algorithm con-verged to the final solution after 330 iterations (100%), ofwhich 49 iterations (15%) are used in Phase I and 281 iter-ations (85%) are used in Phase II. By the end of Phase I,

the objective function increased to 88% of the maximumvalue and the level of non-discreteness decreased to 15%(from an initial value of 44%). In Phase II, the change inthe objective function is relatively smaller than that in PhaseI, nevertheless, it is essential to carefully update the non-linearity variable α to avoid numerical instabilities in thecontinuation approach. The design algorithm converged toa final design withMnd = 1.0%.

Figure 8a shows the final design for the case Rmin = 3&,with the small circle included above the figure indicatingthe filter size. We note that, except for the staircasing inher-ited intrinsically in the FDTD discretiziation, the use of theopen–close filter removed both the metallic (black color)

Fig. 9 a The final conductivitydistribution over the one-layerdesign domain, when we use thetwo-phase filtering approachwith Rmin = 5&. b Thescattering parameters of thetransition

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1774 Emadeldeen Hassan et al.

Fig. 10 a The finalconductivity distribution overthe one-layer design domain,when we use the two-phasefiltering approach withRmin = 7&. b The scatteringparameters of the transition

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and the dielectric (white color) features smaller than the fil-ter size. The performance of the transition, given in Fig.8b, is essentially the same as the one obtained by the linearfilter.

To illustrate the effect of the parameter Rmin, we presenttwo additional designs obtained by the design algorithm forRmin = 5& and Rmin = 7&, in Figs. 9 and 10, which wereobtained after 336 and 296 iterations with a final Mnd =0.7% and Mnd = 1.5%, respectively.

To obtain a quantitative measure of the size control in thefinal designs, we estimate their minimum sizes by seekingthe largest R so that for any edge e in the design there existsa circle Ce with radius R that contains e and is inscribed

in the design. We estimate the minimum size of the metal-lic parts as well as the dielectric protrusions in the finaldesigns. The estimated minimum sizes for the metallic partsof the designs in Figs. 8a, 9a, and 10a are 3&, 6&, and 7&,respectively. The corresponding dielectric protrusions haveestimated minimum sizes 3&, 5&, and 7&, respectively.Therefore, we may conclude that the use of the open–closefilter imposes minimum size control over the metallic partsas well as the dielectric protrusions, with minor impact onthe performance compared to the results obtained with thelinear filter.

Similar to the linear filter case, we use the proposednonlinear filtering to investigate the design of a two-layer

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(c)Fig. 11 Results of optimizing the two-layer design by using the two-phase filtering approach with Rmin = 3&. a First layer conductivitydistribution, z = 1.27 mm. b Second layer conductivity distribution, z = 2.54 mm. c The scattering parameters of the transition

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(c)Fig. 12 Results of optimizing the two-layer design by using the two-phase filtering approach with Rmin = 5&. a First layer conductivitydistribution, z = 1.27 mm. b Second layer conductivity distribution, z = 2.54 mm. c The scattering parameters of the transition

transition. Figures 11, 12, and 13 show the optimizationresults obtained by the algorithm when we use Rmin =3&, Rmin = 5&, and Rmin = 7&, respectively. Theseresults were obtained by the design algorithm after 334,350, and 316 iterations with final Mnd of 0.3%, 0.3%,and 0.9%, respectively. The effect of the filter radius Rminis more prominent on the second layer of the designs inFigs. 11b, 12b, and 13b, where we see that the small featuresin the final design have a size comparable to the corre-sponding value of Rmin. The estimated minimum size of thefirst layer’s metallic parts are 6&, 11&, and 8& for the

designs in Figs. 11a, 12a, and 13a, respectively, whilethe metallic parts in the second layer (Figs. 11b, 12b,and 13b) have estimated minimum sizes 3&, 5&, and 7&,respectively. The dielectric protrusions in both layers inFigs. 11, 12, and 13 have estimated minimum size 3&, 5&,and 7&, respectively. That is, in all cases and for both themetallic and dielctric parts the estimated minimum sizes aregreater than or equal to the employed filter radii. Overall,the use of the open–close filter imposed minimum size con-trol over the features of the designs with minor impact ondesign performance.

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(c)Fig. 13 Results of optimizing the two-layer design by using the two-phase filtering approach with Rmin = 7&. a First layer conductivitydistribution, z = 1.27 mm. b Second layer conductivity distribution, z = 2.54 mm. c The scattering parameters of the transition

1776 Emadeldeen Hassan et al.

8 Conclusion

The filtering of the design variables is essential to counter-act the self penalization of the optimization problem and toavoid designs with poor performance. The simplest use offiltering, that is, a linear filter combined with a continuationapproach over the filter radius, makes it possible to combatthe self penalization issue; however, undesirable small fea-tures sometimes appear in the final design. To control thesize of the small features in the design, we propose a two-phase filtering approach, based on a cascade of fW-mean fil-ters. The numerical experiments, cross-verified with a com-mercial solver, indicate that the proposed filtering approachmakes it possible to impose minimum size control withminor impact on the performance of the designs.

Acknowledgments This work is financially supported by theSwedish Foundation for Strategic Research (No. AM13-0029), theSwedish Research Council (No. 621-3706), and by the Swedish strate-gic research program eSSENCE. The computations were performedon resources provided by the Swedish National Infrastructure forComputing (SNIC) at LUNARC and HPC2N centers.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes were made.

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Topology optimization of compact wideband coaxial-to-waveguide transitions...AbstractIntroductionProblem statementGoverning equations and discretizationTopology optimization problemFilteringLinear filterNon-linear filter

Design algorithmResults and discussionLinear filterNonlinear filters

ConclusionAcknowledgmentsOpen AccessReferences